Appendix: On the Reduction of Syllogisms

Appendix — On the Reduction of the Syllogism

To “reduce” a syllogism means to restate it in the shape of the First Figure. Hence only syllogisms of the Second, Third, and Fourth Figure are reducible. The First Figure has the following four moods:

AAA    EAE    AII    EIO

Examples:

1. (AAA)

All men are mortal John is a man Therefore, John is mortal.

2. (EAE)

No man is a spirit John is a man Therefore, John is not a spirit.

3. (AII)

All men are mortal Some rational beings are men Therefore some rational beings are mortal.

4. (EIO)

No man is a spirit Some rational beings are men Therefore, some rational beings are not spirits.

NOTE: It will be recalled from the Chapter on the classification of propositions that the singular proposition is equivalent to the universal proposition, since the definition of the universal proposition is that it takes its subject in full Extension; and the singular proposition meets this requirement. This note is added, lest the student be surprised to find such a proposition as “John is a man” listed as an A-proposition.

Now, the AAA mood of the First Figure is indicated by a proper name which contains all these vowels. It is called “Barbara.” The EAE mood of the First Figure is called, for like reason, “Celarent.” The AII mood of the First Figure is called “Darii.” The EIO mood of the First Figure is called “Ferio.”

In like manner all the moods of the other Figures (Second, Third, Fourth) are given proper names, and the whole is set forth in a Latin mnemonic stanza. Many of the consonants in the names have meanings which we shall presently explain. The mnemonic is:

Barbara, Celarent, Darii, Ferio, sunt prioris; (I Figure) Cesare, Camestres, Festino, Baroco, secundae; (II Figure) Tertia: Darapti, Disamis, Datisi, Felapton, Bocardo, Ferison, habet; (III Figure) Quarta insuper addit: Bramantip, Camenes, Dimaris, Fesapo, Fresison. (IV Figure)

Now all the moods of the Second, Third, and Fourth Figures cannot be indiscriminately reduced to any desired mood of the First; but in each case the mood of the First Figure to which reduction is to be made will be determined by the mood and figure of the syllogism to be reduced. Therefore, in taking up any syllogism with the purpose of reducing it, the first thing to determine is the figure and the mood of this syllogism as it stands.

To identify the figure of a syllogism, look to the position of the middle term in the premisses. If the middle term is:

1. subject of major premiss, predicate of minor — syllogism is in the First Figure;
2. predicate of both premisses — syllogism is in the Second Figure;
3. subject of both premisses — syllogism is in the Third Figure;
4. predicate of major premiss, subject of minor — syllogism is in the Fourth Figure.

To identify the mood, look for the valid moods in the mnemonic line which indicates the figure already identified. Find the word (proper name) which has, in order, the vowels which stand for the propositions of the syllogism to be reduced.

Worked Example 1

Suppose we have this syllogism to reduce:

All men are mortal beings (A-proposition) Some men are wise beings (I-proposition) Therefore some wise beings are mortal beings (I-proposition)

First, we find that the middle term men is subject of both premisses. Thus we know that the syllogism is in the Third Figure.

Next, we take up the mnemonic line for the Third Figure: Darapti, Disamis, Datisi, Felapton, Bocardo, Ferison, and we look for the word which has the vowels (A-I-I) which stand for the propositions of the syllogism to be reduced, in order. The word “Datisi” contains these vowels in order. Thus we discover that our syllogism is in the Third Figure in Datisi. Now, to reduce it:

1. The syllogism is in Datisi, a name with the initial D. This means that it can be reduced only to that mood of the First Figure which begins in D, that is, to Darii.
2. Next we look for the consonants in the name Datisi. For in the mnemonic names certain consonants have value:
   • s means that the premiss designated by the vowel preceding is to be converted simply;
   • p means that the premiss designated by the vowel preceding is to be converted per accidens, or accidentally;
   • m means that the major and minor premisses are to change places;
   • c means that conversion is to be indirect.

Now, in Datisi we find the letter “s” following the vowel which indicates the minor premiss. This premiss then is to be converted simply. No other change is indicated. We make the reduction as follows:

All men are mortal beings Some wise beings are men Therefore, some wise beings are mortal beings.

Worked Example 2

Take another example:

(A) All men are mortal beings (A) All mortal beings are bodily beings (I) Therefore, some bodily beings are men.

To reduce:

1. The figure is the Fourth, for the middle term is the predicate of the major premiss and the subject of the minor.
2. The mood is Bramantip, for this is the word in the mnemonic line for the Fourth Figure which contains in order the vowels which designate the propositions of the syllogism (A-A-I). We conclude that reduction must be made to Barbara, since Bramantip begins with B.
3. We find the significant consonants “m” and “p” in Bramantip. The consonant “m” tells us to transpose the premisses, making them “change places.” “p,” since it refers to the conclusion and not to a premiss, may be ignored.
4. Making the reduction we have:

(A) All mortal beings are bodily beings (A) All men are mortal beings (A) Therefore, all men are bodily beings.

The student is to note that syllogisms of the Second Figure in Baroco, and those in the Third Figure in Bocardo, cannot be reduced directly, since they contain O-propositions and, as we have learned, these are not directly convertible.